L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s − 0.645·11-s + 4.64·13-s + 16-s + 7.29·17-s − 6.29·19-s + 20-s − 0.645·22-s + 3·23-s + 25-s + 4.64·26-s − 2·31-s + 32-s + 7.29·34-s + 9.93·37-s − 6.29·38-s + 40-s − 6.64·41-s + 0.708·43-s − 0.645·44-s + 3·46-s − 11.5·47-s + 50-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 0.194·11-s + 1.28·13-s + 0.250·16-s + 1.76·17-s − 1.44·19-s + 0.223·20-s − 0.137·22-s + 0.625·23-s + 0.200·25-s + 0.911·26-s − 0.359·31-s + 0.176·32-s + 1.25·34-s + 1.63·37-s − 1.02·38-s + 0.158·40-s − 1.03·41-s + 0.108·43-s − 0.0973·44-s + 0.442·46-s − 1.68·47-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.781802940\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.781802940\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 0.645T + 11T^{2} \) |
| 13 | \( 1 - 4.64T + 13T^{2} \) |
| 17 | \( 1 - 7.29T + 17T^{2} \) |
| 19 | \( 1 + 6.29T + 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 9.93T + 37T^{2} \) |
| 41 | \( 1 + 6.64T + 41T^{2} \) |
| 43 | \( 1 - 0.708T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 1.29T + 59T^{2} \) |
| 61 | \( 1 + 0.708T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 0.708T + 79T^{2} \) |
| 83 | \( 1 + 4.70T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.302643757545121043643874727279, −7.60785134022735543321388028261, −6.65634841306899133057899308713, −6.08103058325830338523744186826, −5.47791341598228240602591695787, −4.65585427305116763052514775733, −3.73758853955122167142797829849, −3.09404299053063156529672327444, −2.02961425548103515255595444072, −1.04695850267207870490407031966,
1.04695850267207870490407031966, 2.02961425548103515255595444072, 3.09404299053063156529672327444, 3.73758853955122167142797829849, 4.65585427305116763052514775733, 5.47791341598228240602591695787, 6.08103058325830338523744186826, 6.65634841306899133057899308713, 7.60785134022735543321388028261, 8.302643757545121043643874727279